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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math.stat.inference;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math.MathException;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math.stat.descriptive.StatisticalSummary;<a name="line.20"></a>
<FONT color="green">021</FONT>    <a name="line.21"></a>
<FONT color="green">022</FONT>    /**<a name="line.22"></a>
<FONT color="green">023</FONT>     * An interface for Student's t-tests.<a name="line.23"></a>
<FONT color="green">024</FONT>     * &lt;p&gt;<a name="line.24"></a>
<FONT color="green">025</FONT>     * Tests can be:&lt;ul&gt;<a name="line.25"></a>
<FONT color="green">026</FONT>     * &lt;li&gt;One-sample or two-sample&lt;/li&gt;<a name="line.26"></a>
<FONT color="green">027</FONT>     * &lt;li&gt;One-sided or two-sided&lt;/li&gt;<a name="line.27"></a>
<FONT color="green">028</FONT>     * &lt;li&gt;Paired or unpaired (for two-sample tests)&lt;/li&gt;<a name="line.28"></a>
<FONT color="green">029</FONT>     * &lt;li&gt;Homoscedastic (equal variance assumption) or heteroscedastic<a name="line.29"></a>
<FONT color="green">030</FONT>     * (for two sample tests)&lt;/li&gt;<a name="line.30"></a>
<FONT color="green">031</FONT>     * &lt;li&gt;Fixed significance level (boolean-valued) or returning p-values.<a name="line.31"></a>
<FONT color="green">032</FONT>     * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.32"></a>
<FONT color="green">033</FONT>     * &lt;p&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     * Test statistics are available for all tests.  Methods including "Test" in<a name="line.34"></a>
<FONT color="green">035</FONT>     * in their names perform tests, all other methods return t-statistics.  Among<a name="line.35"></a>
<FONT color="green">036</FONT>     * the "Test" methods, &lt;code&gt;double-&lt;/code&gt;valued methods return p-values;<a name="line.36"></a>
<FONT color="green">037</FONT>     * &lt;code&gt;boolean-&lt;/code&gt;valued methods perform fixed significance level tests.<a name="line.37"></a>
<FONT color="green">038</FONT>     * Significance levels are always specified as numbers between 0 and 0.5<a name="line.38"></a>
<FONT color="green">039</FONT>     * (e.g. tests at the 95% level  use &lt;code&gt;alpha=0.05&lt;/code&gt;).&lt;/p&gt;<a name="line.39"></a>
<FONT color="green">040</FONT>     * &lt;p&gt;<a name="line.40"></a>
<FONT color="green">041</FONT>     * Input to tests can be either &lt;code&gt;double[]&lt;/code&gt; arrays or <a name="line.41"></a>
<FONT color="green">042</FONT>     * {@link StatisticalSummary} instances.&lt;/p&gt;<a name="line.42"></a>
<FONT color="green">043</FONT>     * <a name="line.43"></a>
<FONT color="green">044</FONT>     *<a name="line.44"></a>
<FONT color="green">045</FONT>     * @version $Revision: 670469 $ $Date: 2008-06-23 04:01:38 -0400 (Mon, 23 Jun 2008) $ <a name="line.45"></a>
<FONT color="green">046</FONT>     */<a name="line.46"></a>
<FONT color="green">047</FONT>    public interface TTest {<a name="line.47"></a>
<FONT color="green">048</FONT>        /**<a name="line.48"></a>
<FONT color="green">049</FONT>         * Computes a paired, 2-sample t-statistic based on the data in the input <a name="line.49"></a>
<FONT color="green">050</FONT>         * arrays.  The t-statistic returned is equivalent to what would be returned by<a name="line.50"></a>
<FONT color="green">051</FONT>         * computing the one-sample t-statistic {@link #t(double, double[])}, with<a name="line.51"></a>
<FONT color="green">052</FONT>         * &lt;code&gt;mu = 0&lt;/code&gt; and the sample array consisting of the (signed) <a name="line.52"></a>
<FONT color="green">053</FONT>         * differences between corresponding entries in &lt;code&gt;sample1&lt;/code&gt; and <a name="line.53"></a>
<FONT color="green">054</FONT>         * &lt;code&gt;sample2.&lt;/code&gt;<a name="line.54"></a>
<FONT color="green">055</FONT>         * &lt;p&gt;<a name="line.55"></a>
<FONT color="green">056</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.56"></a>
<FONT color="green">057</FONT>         * &lt;li&gt;The input arrays must have the same length and their common length<a name="line.57"></a>
<FONT color="green">058</FONT>         * must be at least 2.<a name="line.58"></a>
<FONT color="green">059</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.59"></a>
<FONT color="green">060</FONT>         *<a name="line.60"></a>
<FONT color="green">061</FONT>         * @param sample1 array of sample data values<a name="line.61"></a>
<FONT color="green">062</FONT>         * @param sample2 array of sample data values<a name="line.62"></a>
<FONT color="green">063</FONT>         * @return t statistic<a name="line.63"></a>
<FONT color="green">064</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.64"></a>
<FONT color="green">065</FONT>         * @throws MathException if the statistic can not be computed do to a<a name="line.65"></a>
<FONT color="green">066</FONT>         *         convergence or other numerical error.<a name="line.66"></a>
<FONT color="green">067</FONT>         */<a name="line.67"></a>
<FONT color="green">068</FONT>        public abstract double pairedT(double[] sample1, double[] sample2)<a name="line.68"></a>
<FONT color="green">069</FONT>            throws IllegalArgumentException, MathException;<a name="line.69"></a>
<FONT color="green">070</FONT>        /**<a name="line.70"></a>
<FONT color="green">071</FONT>         * Returns the &lt;i&gt;observed significance level&lt;/i&gt;, or <a name="line.71"></a>
<FONT color="green">072</FONT>         * &lt;i&gt; p-value&lt;/i&gt;, associated with a paired, two-sample, two-tailed t-test <a name="line.72"></a>
<FONT color="green">073</FONT>         * based on the data in the input arrays.<a name="line.73"></a>
<FONT color="green">074</FONT>         * &lt;p&gt;<a name="line.74"></a>
<FONT color="green">075</FONT>         * The number returned is the smallest significance level<a name="line.75"></a>
<FONT color="green">076</FONT>         * at which one can reject the null hypothesis that the mean of the paired<a name="line.76"></a>
<FONT color="green">077</FONT>         * differences is 0 in favor of the two-sided alternative that the mean paired <a name="line.77"></a>
<FONT color="green">078</FONT>         * difference is not equal to 0. For a one-sided test, divide the returned <a name="line.78"></a>
<FONT color="green">079</FONT>         * value by 2.&lt;/p&gt;<a name="line.79"></a>
<FONT color="green">080</FONT>         * &lt;p&gt;<a name="line.80"></a>
<FONT color="green">081</FONT>         * This test is equivalent to a one-sample t-test computed using<a name="line.81"></a>
<FONT color="green">082</FONT>         * {@link #tTest(double, double[])} with &lt;code&gt;mu = 0&lt;/code&gt; and the sample<a name="line.82"></a>
<FONT color="green">083</FONT>         * array consisting of the signed differences between corresponding elements of <a name="line.83"></a>
<FONT color="green">084</FONT>         * &lt;code&gt;sample1&lt;/code&gt; and &lt;code&gt;sample2.&lt;/code&gt;&lt;/p&gt;<a name="line.84"></a>
<FONT color="green">085</FONT>         * &lt;p&gt;<a name="line.85"></a>
<FONT color="green">086</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.86"></a>
<FONT color="green">087</FONT>         * The validity of the p-value depends on the assumptions of the parametric<a name="line.87"></a>
<FONT color="green">088</FONT>         * t-test procedure, as discussed <a name="line.88"></a>
<FONT color="green">089</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;<a name="line.89"></a>
<FONT color="green">090</FONT>         * here&lt;/a&gt;&lt;/p&gt;<a name="line.90"></a>
<FONT color="green">091</FONT>         * &lt;p&gt;<a name="line.91"></a>
<FONT color="green">092</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.92"></a>
<FONT color="green">093</FONT>         * &lt;li&gt;The input array lengths must be the same and their common length must<a name="line.93"></a>
<FONT color="green">094</FONT>         * be at least 2.<a name="line.94"></a>
<FONT color="green">095</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.95"></a>
<FONT color="green">096</FONT>         *<a name="line.96"></a>
<FONT color="green">097</FONT>         * @param sample1 array of sample data values<a name="line.97"></a>
<FONT color="green">098</FONT>         * @param sample2 array of sample data values<a name="line.98"></a>
<FONT color="green">099</FONT>         * @return p-value for t-test<a name="line.99"></a>
<FONT color="green">100</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.100"></a>
<FONT color="green">101</FONT>         * @throws MathException if an error occurs computing the p-value<a name="line.101"></a>
<FONT color="green">102</FONT>         */<a name="line.102"></a>
<FONT color="green">103</FONT>        public abstract double pairedTTest(double[] sample1, double[] sample2)<a name="line.103"></a>
<FONT color="green">104</FONT>            throws IllegalArgumentException, MathException;<a name="line.104"></a>
<FONT color="green">105</FONT>        /**<a name="line.105"></a>
<FONT color="green">106</FONT>         * Performs a paired t-test evaluating the null hypothesis that the <a name="line.106"></a>
<FONT color="green">107</FONT>         * mean of the paired differences between &lt;code&gt;sample1&lt;/code&gt; and<a name="line.107"></a>
<FONT color="green">108</FONT>         * &lt;code&gt;sample2&lt;/code&gt; is 0 in favor of the two-sided alternative that the <a name="line.108"></a>
<FONT color="green">109</FONT>         * mean paired difference is not equal to 0, with significance level <a name="line.109"></a>
<FONT color="green">110</FONT>         * &lt;code&gt;alpha&lt;/code&gt;.<a name="line.110"></a>
<FONT color="green">111</FONT>         * &lt;p&gt;<a name="line.111"></a>
<FONT color="green">112</FONT>         * Returns &lt;code&gt;true&lt;/code&gt; iff the null hypothesis can be rejected with <a name="line.112"></a>
<FONT color="green">113</FONT>         * confidence &lt;code&gt;1 - alpha&lt;/code&gt;.  To perform a 1-sided test, use <a name="line.113"></a>
<FONT color="green">114</FONT>         * &lt;code&gt;alpha * 2&lt;/code&gt;&lt;/p&gt;<a name="line.114"></a>
<FONT color="green">115</FONT>         * &lt;p&gt;<a name="line.115"></a>
<FONT color="green">116</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.116"></a>
<FONT color="green">117</FONT>         * The validity of the test depends on the assumptions of the parametric<a name="line.117"></a>
<FONT color="green">118</FONT>         * t-test procedure, as discussed <a name="line.118"></a>
<FONT color="green">119</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;<a name="line.119"></a>
<FONT color="green">120</FONT>         * here&lt;/a&gt;&lt;/p&gt;<a name="line.120"></a>
<FONT color="green">121</FONT>         * &lt;p&gt;<a name="line.121"></a>
<FONT color="green">122</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.122"></a>
<FONT color="green">123</FONT>         * &lt;li&gt;The input array lengths must be the same and their common length <a name="line.123"></a>
<FONT color="green">124</FONT>         * must be at least 2.<a name="line.124"></a>
<FONT color="green">125</FONT>         * &lt;/li&gt;<a name="line.125"></a>
<FONT color="green">126</FONT>         * &lt;li&gt; &lt;code&gt; 0 &lt; alpha &lt; 0.5 &lt;/code&gt;<a name="line.126"></a>
<FONT color="green">127</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.127"></a>
<FONT color="green">128</FONT>         *<a name="line.128"></a>
<FONT color="green">129</FONT>         * @param sample1 array of sample data values<a name="line.129"></a>
<FONT color="green">130</FONT>         * @param sample2 array of sample data values<a name="line.130"></a>
<FONT color="green">131</FONT>         * @param alpha significance level of the test<a name="line.131"></a>
<FONT color="green">132</FONT>         * @return true if the null hypothesis can be rejected with <a name="line.132"></a>
<FONT color="green">133</FONT>         * confidence 1 - alpha<a name="line.133"></a>
<FONT color="green">134</FONT>         * @throws IllegalArgumentException if the preconditions are not met<a name="line.134"></a>
<FONT color="green">135</FONT>         * @throws MathException if an error occurs performing the test<a name="line.135"></a>
<FONT color="green">136</FONT>         */<a name="line.136"></a>
<FONT color="green">137</FONT>        public abstract boolean pairedTTest(<a name="line.137"></a>
<FONT color="green">138</FONT>            double[] sample1,<a name="line.138"></a>
<FONT color="green">139</FONT>            double[] sample2,<a name="line.139"></a>
<FONT color="green">140</FONT>            double alpha)<a name="line.140"></a>
<FONT color="green">141</FONT>            throws IllegalArgumentException, MathException;<a name="line.141"></a>
<FONT color="green">142</FONT>        /**<a name="line.142"></a>
<FONT color="green">143</FONT>         * Computes a &lt;a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm#formula"&gt; <a name="line.143"></a>
<FONT color="green">144</FONT>         * t statistic &lt;/a&gt; given observed values and a comparison constant.<a name="line.144"></a>
<FONT color="green">145</FONT>         * &lt;p&gt;<a name="line.145"></a>
<FONT color="green">146</FONT>         * This statistic can be used to perform a one sample t-test for the mean.<a name="line.146"></a>
<FONT color="green">147</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.147"></a>
<FONT color="green">148</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.148"></a>
<FONT color="green">149</FONT>         * &lt;li&gt;The observed array length must be at least 2.<a name="line.149"></a>
<FONT color="green">150</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.150"></a>
<FONT color="green">151</FONT>         *<a name="line.151"></a>
<FONT color="green">152</FONT>         * @param mu comparison constant<a name="line.152"></a>
<FONT color="green">153</FONT>         * @param observed array of values<a name="line.153"></a>
<FONT color="green">154</FONT>         * @return t statistic<a name="line.154"></a>
<FONT color="green">155</FONT>         * @throws IllegalArgumentException if input array length is less than 2<a name="line.155"></a>
<FONT color="green">156</FONT>         */<a name="line.156"></a>
<FONT color="green">157</FONT>        public abstract double t(double mu, double[] observed)<a name="line.157"></a>
<FONT color="green">158</FONT>            throws IllegalArgumentException;<a name="line.158"></a>
<FONT color="green">159</FONT>        /**<a name="line.159"></a>
<FONT color="green">160</FONT>         * Computes a &lt;a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm#formula"&gt;<a name="line.160"></a>
<FONT color="green">161</FONT>         * t statistic &lt;/a&gt; to use in comparing the mean of the dataset described by <a name="line.161"></a>
<FONT color="green">162</FONT>         * &lt;code&gt;sampleStats&lt;/code&gt; to &lt;code&gt;mu&lt;/code&gt;.<a name="line.162"></a>
<FONT color="green">163</FONT>         * &lt;p&gt;<a name="line.163"></a>
<FONT color="green">164</FONT>         * This statistic can be used to perform a one sample t-test for the mean.<a name="line.164"></a>
<FONT color="green">165</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.165"></a>
<FONT color="green">166</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.166"></a>
<FONT color="green">167</FONT>         * &lt;li&gt;&lt;code&gt;observed.getN() &gt; = 2&lt;/code&gt;.<a name="line.167"></a>
<FONT color="green">168</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.168"></a>
<FONT color="green">169</FONT>         *<a name="line.169"></a>
<FONT color="green">170</FONT>         * @param mu comparison constant<a name="line.170"></a>
<FONT color="green">171</FONT>         * @param sampleStats DescriptiveStatistics holding sample summary statitstics<a name="line.171"></a>
<FONT color="green">172</FONT>         * @return t statistic<a name="line.172"></a>
<FONT color="green">173</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.173"></a>
<FONT color="green">174</FONT>         */<a name="line.174"></a>
<FONT color="green">175</FONT>        public abstract double t(double mu, StatisticalSummary sampleStats)<a name="line.175"></a>
<FONT color="green">176</FONT>            throws IllegalArgumentException;<a name="line.176"></a>
<FONT color="green">177</FONT>        /**<a name="line.177"></a>
<FONT color="green">178</FONT>         * Computes a 2-sample t statistic,  under the hypothesis of equal <a name="line.178"></a>
<FONT color="green">179</FONT>         * subpopulation variances.  To compute a t-statistic without the<a name="line.179"></a>
<FONT color="green">180</FONT>         * equal variances hypothesis, use {@link #t(double[], double[])}.<a name="line.180"></a>
<FONT color="green">181</FONT>         * &lt;p&gt;<a name="line.181"></a>
<FONT color="green">182</FONT>         * This statistic can be used to perform a (homoscedastic) two-sample<a name="line.182"></a>
<FONT color="green">183</FONT>         * t-test to compare sample means.&lt;/p&gt;<a name="line.183"></a>
<FONT color="green">184</FONT>         * &lt;p&gt;<a name="line.184"></a>
<FONT color="green">185</FONT>         * The t-statisitc is&lt;/p&gt;<a name="line.185"></a>
<FONT color="green">186</FONT>         * &lt;p&gt;<a name="line.186"></a>
<FONT color="green">187</FONT>         * &amp;nbsp;&amp;nbsp;&lt;code&gt;  t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))&lt;/code&gt;<a name="line.187"></a>
<FONT color="green">188</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.188"></a>
<FONT color="green">189</FONT>         * where &lt;strong&gt;&lt;code&gt;n1&lt;/code&gt;&lt;/strong&gt; is the size of first sample; <a name="line.189"></a>
<FONT color="green">190</FONT>         * &lt;strong&gt;&lt;code&gt; n2&lt;/code&gt;&lt;/strong&gt; is the size of second sample; <a name="line.190"></a>
<FONT color="green">191</FONT>         * &lt;strong&gt;&lt;code&gt; m1&lt;/code&gt;&lt;/strong&gt; is the mean of first sample;  <a name="line.191"></a>
<FONT color="green">192</FONT>         * &lt;strong&gt;&lt;code&gt; m2&lt;/code&gt;&lt;/strong&gt; is the mean of second sample&lt;/li&gt;<a name="line.192"></a>
<FONT color="green">193</FONT>         * &lt;/ul&gt;<a name="line.193"></a>
<FONT color="green">194</FONT>         * and &lt;strong&gt;&lt;code&gt;var&lt;/code&gt;&lt;/strong&gt; is the pooled variance estimate:<a name="line.194"></a>
<FONT color="green">195</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.195"></a>
<FONT color="green">196</FONT>         * &lt;code&gt;var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))&lt;/code&gt;<a name="line.196"></a>
<FONT color="green">197</FONT>         * &lt;/p&gt;&lt;p&gt; <a name="line.197"></a>
<FONT color="green">198</FONT>         * with &lt;strong&gt;&lt;code&gt;var1&lt;code&gt;&lt;/strong&gt; the variance of the first sample and<a name="line.198"></a>
<FONT color="green">199</FONT>         * &lt;strong&gt;&lt;code&gt;var2&lt;/code&gt;&lt;/strong&gt; the variance of the second sample.<a name="line.199"></a>
<FONT color="green">200</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.200"></a>
<FONT color="green">201</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.201"></a>
<FONT color="green">202</FONT>         * &lt;li&gt;The observed array lengths must both be at least 2.<a name="line.202"></a>
<FONT color="green">203</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.203"></a>
<FONT color="green">204</FONT>         *<a name="line.204"></a>
<FONT color="green">205</FONT>         * @param sample1 array of sample data values<a name="line.205"></a>
<FONT color="green">206</FONT>         * @param sample2 array of sample data values<a name="line.206"></a>
<FONT color="green">207</FONT>         * @return t statistic<a name="line.207"></a>
<FONT color="green">208</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.208"></a>
<FONT color="green">209</FONT>         */<a name="line.209"></a>
<FONT color="green">210</FONT>        public abstract double homoscedasticT(double[] sample1, double[] sample2)<a name="line.210"></a>
<FONT color="green">211</FONT>            throws IllegalArgumentException;<a name="line.211"></a>
<FONT color="green">212</FONT>        /**<a name="line.212"></a>
<FONT color="green">213</FONT>         * Computes a 2-sample t statistic, without the hypothesis of equal<a name="line.213"></a>
<FONT color="green">214</FONT>         * subpopulation variances.  To compute a t-statistic assuming equal<a name="line.214"></a>
<FONT color="green">215</FONT>         * variances, use {@link #homoscedasticT(double[], double[])}.<a name="line.215"></a>
<FONT color="green">216</FONT>         * &lt;p&gt;<a name="line.216"></a>
<FONT color="green">217</FONT>         * This statistic can be used to perform a two-sample t-test to compare<a name="line.217"></a>
<FONT color="green">218</FONT>         * sample means.&lt;/p&gt;<a name="line.218"></a>
<FONT color="green">219</FONT>         * &lt;p&gt;<a name="line.219"></a>
<FONT color="green">220</FONT>         * The t-statisitc is&lt;/p&gt;<a name="line.220"></a>
<FONT color="green">221</FONT>         * &lt;p&gt;<a name="line.221"></a>
<FONT color="green">222</FONT>         * &amp;nbsp;&amp;nbsp; &lt;code&gt;  t = (m1 - m2) / sqrt(var1/n1 + var2/n2)&lt;/code&gt;<a name="line.222"></a>
<FONT color="green">223</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.223"></a>
<FONT color="green">224</FONT>         *  where &lt;strong&gt;&lt;code&gt;n1&lt;/code&gt;&lt;/strong&gt; is the size of the first sample<a name="line.224"></a>
<FONT color="green">225</FONT>         * &lt;strong&gt;&lt;code&gt; n2&lt;/code&gt;&lt;/strong&gt; is the size of the second sample; <a name="line.225"></a>
<FONT color="green">226</FONT>         * &lt;strong&gt;&lt;code&gt; m1&lt;/code&gt;&lt;/strong&gt; is the mean of the first sample;  <a name="line.226"></a>
<FONT color="green">227</FONT>         * &lt;strong&gt;&lt;code&gt; m2&lt;/code&gt;&lt;/strong&gt; is the mean of the second sample;<a name="line.227"></a>
<FONT color="green">228</FONT>         * &lt;strong&gt;&lt;code&gt; var1&lt;/code&gt;&lt;/strong&gt; is the variance of the first sample;<a name="line.228"></a>
<FONT color="green">229</FONT>         * &lt;strong&gt;&lt;code&gt; var2&lt;/code&gt;&lt;/strong&gt; is the variance of the second sample;  <a name="line.229"></a>
<FONT color="green">230</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.230"></a>
<FONT color="green">231</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.231"></a>
<FONT color="green">232</FONT>         * &lt;li&gt;The observed array lengths must both be at least 2.<a name="line.232"></a>
<FONT color="green">233</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.233"></a>
<FONT color="green">234</FONT>         *<a name="line.234"></a>
<FONT color="green">235</FONT>         * @param sample1 array of sample data values<a name="line.235"></a>
<FONT color="green">236</FONT>         * @param sample2 array of sample data values<a name="line.236"></a>
<FONT color="green">237</FONT>         * @return t statistic<a name="line.237"></a>
<FONT color="green">238</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.238"></a>
<FONT color="green">239</FONT>         */<a name="line.239"></a>
<FONT color="green">240</FONT>        public abstract double t(double[] sample1, double[] sample2)<a name="line.240"></a>
<FONT color="green">241</FONT>            throws IllegalArgumentException;<a name="line.241"></a>
<FONT color="green">242</FONT>        /**<a name="line.242"></a>
<FONT color="green">243</FONT>         * Computes a 2-sample t statistic &lt;/a&gt;, comparing the means of the datasets<a name="line.243"></a>
<FONT color="green">244</FONT>         * described by two {@link StatisticalSummary} instances, without the<a name="line.244"></a>
<FONT color="green">245</FONT>         * assumption of equal subpopulation variances.  Use <a name="line.245"></a>
<FONT color="green">246</FONT>         * {@link #homoscedasticT(StatisticalSummary, StatisticalSummary)} to<a name="line.246"></a>
<FONT color="green">247</FONT>         * compute a t-statistic under the equal variances assumption.<a name="line.247"></a>
<FONT color="green">248</FONT>         * &lt;p&gt;<a name="line.248"></a>
<FONT color="green">249</FONT>         * This statistic can be used to perform a two-sample t-test to compare<a name="line.249"></a>
<FONT color="green">250</FONT>         * sample means.&lt;/p&gt;<a name="line.250"></a>
<FONT color="green">251</FONT>         * &lt;p&gt;<a name="line.251"></a>
<FONT color="green">252</FONT>          * The returned  t-statisitc is&lt;/p&gt;<a name="line.252"></a>
<FONT color="green">253</FONT>         * &lt;p&gt;<a name="line.253"></a>
<FONT color="green">254</FONT>         * &amp;nbsp;&amp;nbsp; &lt;code&gt;  t = (m1 - m2) / sqrt(var1/n1 + var2/n2)&lt;/code&gt;<a name="line.254"></a>
<FONT color="green">255</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.255"></a>
<FONT color="green">256</FONT>         * where &lt;strong&gt;&lt;code&gt;n1&lt;/code&gt;&lt;/strong&gt; is the size of the first sample; <a name="line.256"></a>
<FONT color="green">257</FONT>         * &lt;strong&gt;&lt;code&gt; n2&lt;/code&gt;&lt;/strong&gt; is the size of the second sample; <a name="line.257"></a>
<FONT color="green">258</FONT>         * &lt;strong&gt;&lt;code&gt; m1&lt;/code&gt;&lt;/strong&gt; is the mean of the first sample;  <a name="line.258"></a>
<FONT color="green">259</FONT>         * &lt;strong&gt;&lt;code&gt; m2&lt;/code&gt;&lt;/strong&gt; is the mean of the second sample<a name="line.259"></a>
<FONT color="green">260</FONT>         * &lt;strong&gt;&lt;code&gt; var1&lt;/code&gt;&lt;/strong&gt; is the variance of the first sample;  <a name="line.260"></a>
<FONT color="green">261</FONT>         * &lt;strong&gt;&lt;code&gt; var2&lt;/code&gt;&lt;/strong&gt; is the variance of the second sample<a name="line.261"></a>
<FONT color="green">262</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.262"></a>
<FONT color="green">263</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.263"></a>
<FONT color="green">264</FONT>         * &lt;li&gt;The datasets described by the two Univariates must each contain<a name="line.264"></a>
<FONT color="green">265</FONT>         * at least 2 observations.<a name="line.265"></a>
<FONT color="green">266</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.266"></a>
<FONT color="green">267</FONT>         *<a name="line.267"></a>
<FONT color="green">268</FONT>         * @param sampleStats1 StatisticalSummary describing data from the first sample<a name="line.268"></a>
<FONT color="green">269</FONT>         * @param sampleStats2 StatisticalSummary describing data from the second sample<a name="line.269"></a>
<FONT color="green">270</FONT>         * @return t statistic<a name="line.270"></a>
<FONT color="green">271</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.271"></a>
<FONT color="green">272</FONT>         */<a name="line.272"></a>
<FONT color="green">273</FONT>        public abstract double t(<a name="line.273"></a>
<FONT color="green">274</FONT>            StatisticalSummary sampleStats1,<a name="line.274"></a>
<FONT color="green">275</FONT>            StatisticalSummary sampleStats2)<a name="line.275"></a>
<FONT color="green">276</FONT>            throws IllegalArgumentException;<a name="line.276"></a>
<FONT color="green">277</FONT>        /**<a name="line.277"></a>
<FONT color="green">278</FONT>         * Computes a 2-sample t statistic, comparing the means of the datasets<a name="line.278"></a>
<FONT color="green">279</FONT>         * described by two {@link StatisticalSummary} instances, under the<a name="line.279"></a>
<FONT color="green">280</FONT>         * assumption of equal subpopulation variances.  To compute a t-statistic<a name="line.280"></a>
<FONT color="green">281</FONT>         * without the equal variances assumption, use <a name="line.281"></a>
<FONT color="green">282</FONT>         * {@link #t(StatisticalSummary, StatisticalSummary)}.<a name="line.282"></a>
<FONT color="green">283</FONT>         * &lt;p&gt;<a name="line.283"></a>
<FONT color="green">284</FONT>         * This statistic can be used to perform a (homoscedastic) two-sample<a name="line.284"></a>
<FONT color="green">285</FONT>         * t-test to compare sample means.&lt;/p&gt;<a name="line.285"></a>
<FONT color="green">286</FONT>         * &lt;p&gt;<a name="line.286"></a>
<FONT color="green">287</FONT>         * The t-statisitc returned is&lt;/p&gt;<a name="line.287"></a>
<FONT color="green">288</FONT>         * &lt;p&gt;<a name="line.288"></a>
<FONT color="green">289</FONT>         * &amp;nbsp;&amp;nbsp;&lt;code&gt;  t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))&lt;/code&gt;<a name="line.289"></a>
<FONT color="green">290</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.290"></a>
<FONT color="green">291</FONT>         * where &lt;strong&gt;&lt;code&gt;n1&lt;/code&gt;&lt;/strong&gt; is the size of first sample; <a name="line.291"></a>
<FONT color="green">292</FONT>         * &lt;strong&gt;&lt;code&gt; n2&lt;/code&gt;&lt;/strong&gt; is the size of second sample; <a name="line.292"></a>
<FONT color="green">293</FONT>         * &lt;strong&gt;&lt;code&gt; m1&lt;/code&gt;&lt;/strong&gt; is the mean of first sample;  <a name="line.293"></a>
<FONT color="green">294</FONT>         * &lt;strong&gt;&lt;code&gt; m2&lt;/code&gt;&lt;/strong&gt; is the mean of second sample<a name="line.294"></a>
<FONT color="green">295</FONT>         * and &lt;strong&gt;&lt;code&gt;var&lt;/code&gt;&lt;/strong&gt; is the pooled variance estimate:<a name="line.295"></a>
<FONT color="green">296</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.296"></a>
<FONT color="green">297</FONT>         * &lt;code&gt;var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))&lt;/code&gt;<a name="line.297"></a>
<FONT color="green">298</FONT>         * &lt;/p&gt;&lt;p&gt; <a name="line.298"></a>
<FONT color="green">299</FONT>         * with &lt;strong&gt;&lt;code&gt;var1&lt;code&gt;&lt;/strong&gt; the variance of the first sample and<a name="line.299"></a>
<FONT color="green">300</FONT>         * &lt;strong&gt;&lt;code&gt;var2&lt;/code&gt;&lt;/strong&gt; the variance of the second sample.<a name="line.300"></a>
<FONT color="green">301</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.301"></a>
<FONT color="green">302</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.302"></a>
<FONT color="green">303</FONT>         * &lt;li&gt;The datasets described by the two Univariates must each contain<a name="line.303"></a>
<FONT color="green">304</FONT>         * at least 2 observations.<a name="line.304"></a>
<FONT color="green">305</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.305"></a>
<FONT color="green">306</FONT>         *<a name="line.306"></a>
<FONT color="green">307</FONT>         * @param sampleStats1 StatisticalSummary describing data from the first sample<a name="line.307"></a>
<FONT color="green">308</FONT>         * @param sampleStats2 StatisticalSummary describing data from the second sample<a name="line.308"></a>
<FONT color="green">309</FONT>         * @return t statistic<a name="line.309"></a>
<FONT color="green">310</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.310"></a>
<FONT color="green">311</FONT>         */<a name="line.311"></a>
<FONT color="green">312</FONT>        public abstract double homoscedasticT(<a name="line.312"></a>
<FONT color="green">313</FONT>            StatisticalSummary sampleStats1,<a name="line.313"></a>
<FONT color="green">314</FONT>            StatisticalSummary sampleStats2)<a name="line.314"></a>
<FONT color="green">315</FONT>            throws IllegalArgumentException;<a name="line.315"></a>
<FONT color="green">316</FONT>        /**<a name="line.316"></a>
<FONT color="green">317</FONT>         * Returns the &lt;i&gt;observed significance level&lt;/i&gt;, or <a name="line.317"></a>
<FONT color="green">318</FONT>         * &lt;i&gt;p-value&lt;/i&gt;, associated with a one-sample, two-tailed t-test <a name="line.318"></a>
<FONT color="green">319</FONT>         * comparing the mean of the input array with the constant &lt;code&gt;mu&lt;/code&gt;.<a name="line.319"></a>
<FONT color="green">320</FONT>         * &lt;p&gt;<a name="line.320"></a>
<FONT color="green">321</FONT>         * The number returned is the smallest significance level<a name="line.321"></a>
<FONT color="green">322</FONT>         * at which one can reject the null hypothesis that the mean equals <a name="line.322"></a>
<FONT color="green">323</FONT>         * &lt;code&gt;mu&lt;/code&gt; in favor of the two-sided alternative that the mean<a name="line.323"></a>
<FONT color="green">324</FONT>         * is different from &lt;code&gt;mu&lt;/code&gt;. For a one-sided test, divide the <a name="line.324"></a>
<FONT color="green">325</FONT>         * returned value by 2.&lt;/p&gt;<a name="line.325"></a>
<FONT color="green">326</FONT>         * &lt;p&gt;<a name="line.326"></a>
<FONT color="green">327</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.327"></a>
<FONT color="green">328</FONT>         * The validity of the test depends on the assumptions of the parametric<a name="line.328"></a>
<FONT color="green">329</FONT>         * t-test procedure, as discussed <a name="line.329"></a>
<FONT color="green">330</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;here&lt;/a&gt;<a name="line.330"></a>
<FONT color="green">331</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.331"></a>
<FONT color="green">332</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.332"></a>
<FONT color="green">333</FONT>         * &lt;li&gt;The observed array length must be at least 2.<a name="line.333"></a>
<FONT color="green">334</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.334"></a>
<FONT color="green">335</FONT>         *<a name="line.335"></a>
<FONT color="green">336</FONT>         * @param mu constant value to compare sample mean against<a name="line.336"></a>
<FONT color="green">337</FONT>         * @param sample array of sample data values<a name="line.337"></a>
<FONT color="green">338</FONT>         * @return p-value<a name="line.338"></a>
<FONT color="green">339</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.339"></a>
<FONT color="green">340</FONT>         * @throws MathException if an error occurs computing the p-value<a name="line.340"></a>
<FONT color="green">341</FONT>         */<a name="line.341"></a>
<FONT color="green">342</FONT>        public abstract double tTest(double mu, double[] sample)<a name="line.342"></a>
<FONT color="green">343</FONT>            throws IllegalArgumentException, MathException;<a name="line.343"></a>
<FONT color="green">344</FONT>        /**<a name="line.344"></a>
<FONT color="green">345</FONT>         * Performs a &lt;a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"&gt;<a name="line.345"></a>
<FONT color="green">346</FONT>         * two-sided t-test&lt;/a&gt; evaluating the null hypothesis that the mean of the population from<a name="line.346"></a>
<FONT color="green">347</FONT>         * which &lt;code&gt;sample&lt;/code&gt; is drawn equals &lt;code&gt;mu&lt;/code&gt;.<a name="line.347"></a>
<FONT color="green">348</FONT>         * &lt;p&gt;<a name="line.348"></a>
<FONT color="green">349</FONT>         * Returns &lt;code&gt;true&lt;/code&gt; iff the null hypothesis can be <a name="line.349"></a>
<FONT color="green">350</FONT>         * rejected with confidence &lt;code&gt;1 - alpha&lt;/code&gt;.  To <a name="line.350"></a>
<FONT color="green">351</FONT>         * perform a 1-sided test, use &lt;code&gt;alpha * 2&lt;/code&gt;&lt;/p&gt;<a name="line.351"></a>
<FONT color="green">352</FONT>         * &lt;p&gt;<a name="line.352"></a>
<FONT color="green">353</FONT>         * &lt;strong&gt;Examples:&lt;/strong&gt;&lt;br&gt;&lt;ol&gt;<a name="line.353"></a>
<FONT color="green">354</FONT>         * &lt;li&gt;To test the (2-sided) hypothesis &lt;code&gt;sample mean = mu &lt;/code&gt; at<a name="line.354"></a>
<FONT color="green">355</FONT>         * the 95% level, use &lt;br&gt;&lt;code&gt;tTest(mu, sample, 0.05) &lt;/code&gt;<a name="line.355"></a>
<FONT color="green">356</FONT>         * &lt;/li&gt;<a name="line.356"></a>
<FONT color="green">357</FONT>         * &lt;li&gt;To test the (one-sided) hypothesis &lt;code&gt; sample mean &lt; mu &lt;/code&gt;<a name="line.357"></a>
<FONT color="green">358</FONT>         * at the 99% level, first verify that the measured sample mean is less <a name="line.358"></a>
<FONT color="green">359</FONT>         * than &lt;code&gt;mu&lt;/code&gt; and then use <a name="line.359"></a>
<FONT color="green">360</FONT>         * &lt;br&gt;&lt;code&gt;tTest(mu, sample, 0.02) &lt;/code&gt;<a name="line.360"></a>
<FONT color="green">361</FONT>         * &lt;/li&gt;&lt;/ol&gt;&lt;/p&gt;<a name="line.361"></a>
<FONT color="green">362</FONT>         * &lt;p&gt;<a name="line.362"></a>
<FONT color="green">363</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.363"></a>
<FONT color="green">364</FONT>         * The validity of the test depends on the assumptions of the one-sample <a name="line.364"></a>
<FONT color="green">365</FONT>         * parametric t-test procedure, as discussed <a name="line.365"></a>
<FONT color="green">366</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/sg_glos.html#one-sample"&gt;here&lt;/a&gt;<a name="line.366"></a>
<FONT color="green">367</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.367"></a>
<FONT color="green">368</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.368"></a>
<FONT color="green">369</FONT>         * &lt;li&gt;The observed array length must be at least 2.<a name="line.369"></a>
<FONT color="green">370</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.370"></a>
<FONT color="green">371</FONT>         *<a name="line.371"></a>
<FONT color="green">372</FONT>         * @param mu constant value to compare sample mean against<a name="line.372"></a>
<FONT color="green">373</FONT>         * @param sample array of sample data values<a name="line.373"></a>
<FONT color="green">374</FONT>         * @param alpha significance level of the test<a name="line.374"></a>
<FONT color="green">375</FONT>         * @return p-value<a name="line.375"></a>
<FONT color="green">376</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.376"></a>
<FONT color="green">377</FONT>         * @throws MathException if an error computing the p-value<a name="line.377"></a>
<FONT color="green">378</FONT>         */<a name="line.378"></a>
<FONT color="green">379</FONT>        public abstract boolean tTest(double mu, double[] sample, double alpha)<a name="line.379"></a>
<FONT color="green">380</FONT>            throws IllegalArgumentException, MathException;<a name="line.380"></a>
<FONT color="green">381</FONT>        /**<a name="line.381"></a>
<FONT color="green">382</FONT>         * Returns the &lt;i&gt;observed significance level&lt;/i&gt;, or <a name="line.382"></a>
<FONT color="green">383</FONT>         * &lt;i&gt;p-value&lt;/i&gt;, associated with a one-sample, two-tailed t-test <a name="line.383"></a>
<FONT color="green">384</FONT>         * comparing the mean of the dataset described by &lt;code&gt;sampleStats&lt;/code&gt;<a name="line.384"></a>
<FONT color="green">385</FONT>         * with the constant &lt;code&gt;mu&lt;/code&gt;.<a name="line.385"></a>
<FONT color="green">386</FONT>         * &lt;p&gt;<a name="line.386"></a>
<FONT color="green">387</FONT>         * The number returned is the smallest significance level<a name="line.387"></a>
<FONT color="green">388</FONT>         * at which one can reject the null hypothesis that the mean equals <a name="line.388"></a>
<FONT color="green">389</FONT>         * &lt;code&gt;mu&lt;/code&gt; in favor of the two-sided alternative that the mean<a name="line.389"></a>
<FONT color="green">390</FONT>         * is different from &lt;code&gt;mu&lt;/code&gt;. For a one-sided test, divide the <a name="line.390"></a>
<FONT color="green">391</FONT>         * returned value by 2.&lt;/p&gt;<a name="line.391"></a>
<FONT color="green">392</FONT>         * &lt;p&gt;<a name="line.392"></a>
<FONT color="green">393</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.393"></a>
<FONT color="green">394</FONT>         * The validity of the test depends on the assumptions of the parametric<a name="line.394"></a>
<FONT color="green">395</FONT>         * t-test procedure, as discussed <a name="line.395"></a>
<FONT color="green">396</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;<a name="line.396"></a>
<FONT color="green">397</FONT>         * here&lt;/a&gt;&lt;/p&gt;<a name="line.397"></a>
<FONT color="green">398</FONT>         * &lt;p&gt;<a name="line.398"></a>
<FONT color="green">399</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.399"></a>
<FONT color="green">400</FONT>         * &lt;li&gt;The sample must contain at least 2 observations.<a name="line.400"></a>
<FONT color="green">401</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.401"></a>
<FONT color="green">402</FONT>         *<a name="line.402"></a>
<FONT color="green">403</FONT>         * @param mu constant value to compare sample mean against<a name="line.403"></a>
<FONT color="green">404</FONT>         * @param sampleStats StatisticalSummary describing sample data<a name="line.404"></a>
<FONT color="green">405</FONT>         * @return p-value<a name="line.405"></a>
<FONT color="green">406</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.406"></a>
<FONT color="green">407</FONT>         * @throws MathException if an error occurs computing the p-value<a name="line.407"></a>
<FONT color="green">408</FONT>         */<a name="line.408"></a>
<FONT color="green">409</FONT>        public abstract double tTest(double mu, StatisticalSummary sampleStats)<a name="line.409"></a>
<FONT color="green">410</FONT>            throws IllegalArgumentException, MathException;<a name="line.410"></a>
<FONT color="green">411</FONT>        /**<a name="line.411"></a>
<FONT color="green">412</FONT>         * Performs a &lt;a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"&gt;<a name="line.412"></a>
<FONT color="green">413</FONT>         * two-sided t-test&lt;/a&gt; evaluating the null hypothesis that the mean of the<a name="line.413"></a>
<FONT color="green">414</FONT>         * population from which the dataset described by &lt;code&gt;stats&lt;/code&gt; is<a name="line.414"></a>
<FONT color="green">415</FONT>         * drawn equals &lt;code&gt;mu&lt;/code&gt;.<a name="line.415"></a>
<FONT color="green">416</FONT>         * &lt;p&gt;<a name="line.416"></a>
<FONT color="green">417</FONT>         * Returns &lt;code&gt;true&lt;/code&gt; iff the null hypothesis can be rejected with<a name="line.417"></a>
<FONT color="green">418</FONT>         * confidence &lt;code&gt;1 - alpha&lt;/code&gt;.  To  perform a 1-sided test, use<a name="line.418"></a>
<FONT color="green">419</FONT>         * &lt;code&gt;alpha * 2.&lt;/code&gt;&lt;/p&gt;<a name="line.419"></a>
<FONT color="green">420</FONT>         * &lt;p&gt;<a name="line.420"></a>
<FONT color="green">421</FONT>         * &lt;strong&gt;Examples:&lt;/strong&gt;&lt;br&gt;&lt;ol&gt;<a name="line.421"></a>
<FONT color="green">422</FONT>         * &lt;li&gt;To test the (2-sided) hypothesis &lt;code&gt;sample mean = mu &lt;/code&gt; at<a name="line.422"></a>
<FONT color="green">423</FONT>         * the 95% level, use &lt;br&gt;&lt;code&gt;tTest(mu, sampleStats, 0.05) &lt;/code&gt;<a name="line.423"></a>
<FONT color="green">424</FONT>         * &lt;/li&gt;<a name="line.424"></a>
<FONT color="green">425</FONT>         * &lt;li&gt;To test the (one-sided) hypothesis &lt;code&gt; sample mean &lt; mu &lt;/code&gt;<a name="line.425"></a>
<FONT color="green">426</FONT>         * at the 99% level, first verify that the measured sample mean is less <a name="line.426"></a>
<FONT color="green">427</FONT>         * than &lt;code&gt;mu&lt;/code&gt; and then use <a name="line.427"></a>
<FONT color="green">428</FONT>         * &lt;br&gt;&lt;code&gt;tTest(mu, sampleStats, 0.02) &lt;/code&gt;<a name="line.428"></a>
<FONT color="green">429</FONT>         * &lt;/li&gt;&lt;/ol&gt;&lt;/p&gt;<a name="line.429"></a>
<FONT color="green">430</FONT>         * &lt;p&gt;<a name="line.430"></a>
<FONT color="green">431</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.431"></a>
<FONT color="green">432</FONT>         * The validity of the test depends on the assumptions of the one-sample <a name="line.432"></a>
<FONT color="green">433</FONT>         * parametric t-test procedure, as discussed <a name="line.433"></a>
<FONT color="green">434</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/sg_glos.html#one-sample"&gt;here&lt;/a&gt;<a name="line.434"></a>
<FONT color="green">435</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.435"></a>
<FONT color="green">436</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.436"></a>
<FONT color="green">437</FONT>         * &lt;li&gt;The sample must include at least 2 observations.<a name="line.437"></a>
<FONT color="green">438</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.438"></a>
<FONT color="green">439</FONT>         *<a name="line.439"></a>
<FONT color="green">440</FONT>         * @param mu constant value to compare sample mean against<a name="line.440"></a>
<FONT color="green">441</FONT>         * @param sampleStats StatisticalSummary describing sample data values<a name="line.441"></a>
<FONT color="green">442</FONT>         * @param alpha significance level of the test<a name="line.442"></a>
<FONT color="green">443</FONT>         * @return p-value<a name="line.443"></a>
<FONT color="green">444</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.444"></a>
<FONT color="green">445</FONT>         * @throws MathException if an error occurs computing the p-value<a name="line.445"></a>
<FONT color="green">446</FONT>         */<a name="line.446"></a>
<FONT color="green">447</FONT>        public abstract boolean tTest(<a name="line.447"></a>
<FONT color="green">448</FONT>            double mu,<a name="line.448"></a>
<FONT color="green">449</FONT>            StatisticalSummary sampleStats,<a name="line.449"></a>
<FONT color="green">450</FONT>            double alpha)<a name="line.450"></a>
<FONT color="green">451</FONT>            throws IllegalArgumentException, MathException;<a name="line.451"></a>
<FONT color="green">452</FONT>        /**<a name="line.452"></a>
<FONT color="green">453</FONT>         * Returns the &lt;i&gt;observed significance level&lt;/i&gt;, or <a name="line.453"></a>
<FONT color="green">454</FONT>         * &lt;i&gt;p-value&lt;/i&gt;, associated with a two-sample, two-tailed t-test <a name="line.454"></a>
<FONT color="green">455</FONT>         * comparing the means of the input arrays.<a name="line.455"></a>
<FONT color="green">456</FONT>         * &lt;p&gt;<a name="line.456"></a>
<FONT color="green">457</FONT>         * The number returned is the smallest significance level<a name="line.457"></a>
<FONT color="green">458</FONT>         * at which one can reject the null hypothesis that the two means are<a name="line.458"></a>
<FONT color="green">459</FONT>         * equal in favor of the two-sided alternative that they are different. <a name="line.459"></a>
<FONT color="green">460</FONT>         * For a one-sided test, divide the returned value by 2.&lt;/p&gt;<a name="line.460"></a>
<FONT color="green">461</FONT>         * &lt;p&gt;<a name="line.461"></a>
<FONT color="green">462</FONT>         * The test does not assume that the underlying popuation variances are<a name="line.462"></a>
<FONT color="green">463</FONT>         * equal  and it uses approximated degrees of freedom computed from the <a name="line.463"></a>
<FONT color="green">464</FONT>         * sample data to compute the p-value.  The t-statistic used is as defined in<a name="line.464"></a>
<FONT color="green">465</FONT>         * {@link #t(double[], double[])} and the Welch-Satterthwaite approximation<a name="line.465"></a>
<FONT color="green">466</FONT>         * to the degrees of freedom is used, <a name="line.466"></a>
<FONT color="green">467</FONT>         * as described <a name="line.467"></a>
<FONT color="green">468</FONT>         * &lt;a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm"&gt;<a name="line.468"></a>
<FONT color="green">469</FONT>         * here.&lt;/a&gt;  To perform the test under the assumption of equal subpopulation<a name="line.469"></a>
<FONT color="green">470</FONT>         * variances, use {@link #homoscedasticTTest(double[], double[])}.&lt;/p&gt;<a name="line.470"></a>
<FONT color="green">471</FONT>         * &lt;p&gt;<a name="line.471"></a>
<FONT color="green">472</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.472"></a>
<FONT color="green">473</FONT>         * The validity of the p-value depends on the assumptions of the parametric<a name="line.473"></a>
<FONT color="green">474</FONT>         * t-test procedure, as discussed <a name="line.474"></a>
<FONT color="green">475</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;<a name="line.475"></a>
<FONT color="green">476</FONT>         * here&lt;/a&gt;&lt;/p&gt;<a name="line.476"></a>
<FONT color="green">477</FONT>         * &lt;p&gt;<a name="line.477"></a>
<FONT color="green">478</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.478"></a>
<FONT color="green">479</FONT>         * &lt;li&gt;The observed array lengths must both be at least 2.<a name="line.479"></a>
<FONT color="green">480</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.480"></a>
<FONT color="green">481</FONT>         *<a name="line.481"></a>
<FONT color="green">482</FONT>         * @param sample1 array of sample data values<a name="line.482"></a>
<FONT color="green">483</FONT>         * @param sample2 array of sample data values<a name="line.483"></a>
<FONT color="green">484</FONT>         * @return p-value for t-test<a name="line.484"></a>
<FONT color="green">485</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.485"></a>
<FONT color="green">486</FONT>         * @throws MathException if an error occurs computing the p-value<a name="line.486"></a>
<FONT color="green">487</FONT>         */<a name="line.487"></a>
<FONT color="green">488</FONT>        public abstract double tTest(double[] sample1, double[] sample2)<a name="line.488"></a>
<FONT color="green">489</FONT>            throws IllegalArgumentException, MathException;<a name="line.489"></a>
<FONT color="green">490</FONT>        /**<a name="line.490"></a>
<FONT color="green">491</FONT>         * Returns the &lt;i&gt;observed significance level&lt;/i&gt;, or <a name="line.491"></a>
<FONT color="green">492</FONT>         * &lt;i&gt;p-value&lt;/i&gt;, associated with a two-sample, two-tailed t-test <a name="line.492"></a>
<FONT color="green">493</FONT>         * comparing the means of the input arrays, under the assumption that<a name="line.493"></a>
<FONT color="green">494</FONT>         * the two samples are drawn from subpopulations with equal variances.<a name="line.494"></a>
<FONT color="green">495</FONT>         * To perform the test without the equal variances assumption, use<a name="line.495"></a>
<FONT color="green">496</FONT>         * {@link #tTest(double[], double[])}.&lt;/p&gt;<a name="line.496"></a>
<FONT color="green">497</FONT>         * &lt;p&gt;<a name="line.497"></a>
<FONT color="green">498</FONT>         * The number returned is the smallest significance level<a name="line.498"></a>
<FONT color="green">499</FONT>         * at which one can reject the null hypothesis that the two means are<a name="line.499"></a>
<FONT color="green">500</FONT>         * equal in favor of the two-sided alternative that they are different. <a name="line.500"></a>
<FONT color="green">501</FONT>         * For a one-sided test, divide the returned value by 2.&lt;/p&gt;<a name="line.501"></a>
<FONT color="green">502</FONT>         * &lt;p&gt;<a name="line.502"></a>
<FONT color="green">503</FONT>         * A pooled variance estimate is used to compute the t-statistic.  See<a name="line.503"></a>
<FONT color="green">504</FONT>         * {@link #homoscedasticT(double[], double[])}. The sum of the sample sizes<a name="line.504"></a>
<FONT color="green">505</FONT>         * minus 2 is used as the degrees of freedom.&lt;/p&gt;<a name="line.505"></a>
<FONT color="green">506</FONT>         * &lt;p&gt;<a name="line.506"></a>
<FONT color="green">507</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.507"></a>
<FONT color="green">508</FONT>         * The validity of the p-value depends on the assumptions of the parametric<a name="line.508"></a>
<FONT color="green">509</FONT>         * t-test procedure, as discussed <a name="line.509"></a>
<FONT color="green">510</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;<a name="line.510"></a>
<FONT color="green">511</FONT>         * here&lt;/a&gt;&lt;/p&gt;<a name="line.511"></a>
<FONT color="green">512</FONT>         * &lt;p&gt;<a name="line.512"></a>
<FONT color="green">513</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.513"></a>
<FONT color="green">514</FONT>         * &lt;li&gt;The observed array lengths must both be at least 2.<a name="line.514"></a>
<FONT color="green">515</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.515"></a>
<FONT color="green">516</FONT>         *<a name="line.516"></a>
<FONT color="green">517</FONT>         * @param sample1 array of sample data values<a name="line.517"></a>
<FONT color="green">518</FONT>         * @param sample2 array of sample data values<a name="line.518"></a>
<FONT color="green">519</FONT>         * @return p-value for t-test<a name="line.519"></a>
<FONT color="green">520</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.520"></a>
<FONT color="green">521</FONT>         * @throws MathException if an error occurs computing the p-value<a name="line.521"></a>
<FONT color="green">522</FONT>         */<a name="line.522"></a>
<FONT color="green">523</FONT>        public abstract double homoscedasticTTest(<a name="line.523"></a>
<FONT color="green">524</FONT>            double[] sample1,<a name="line.524"></a>
<FONT color="green">525</FONT>            double[] sample2)<a name="line.525"></a>
<FONT color="green">526</FONT>            throws IllegalArgumentException, MathException;<a name="line.526"></a>
<FONT color="green">527</FONT>        /**<a name="line.527"></a>
<FONT color="green">528</FONT>         * Performs a <a name="line.528"></a>
<FONT color="green">529</FONT>         * &lt;a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"&gt;<a name="line.529"></a>
<FONT color="green">530</FONT>         * two-sided t-test&lt;/a&gt; evaluating the null hypothesis that &lt;code&gt;sample1&lt;/code&gt; <a name="line.530"></a>
<FONT color="green">531</FONT>         * and &lt;code&gt;sample2&lt;/code&gt; are drawn from populations with the same mean, <a name="line.531"></a>
<FONT color="green">532</FONT>         * with significance level &lt;code&gt;alpha&lt;/code&gt;.  This test does not assume<a name="line.532"></a>
<FONT color="green">533</FONT>         * that the subpopulation variances are equal.  To perform the test assuming<a name="line.533"></a>
<FONT color="green">534</FONT>         * equal variances, use <a name="line.534"></a>
<FONT color="green">535</FONT>         * {@link #homoscedasticTTest(double[], double[], double)}.<a name="line.535"></a>
<FONT color="green">536</FONT>         * &lt;p&gt;<a name="line.536"></a>
<FONT color="green">537</FONT>         * Returns &lt;code&gt;true&lt;/code&gt; iff the null hypothesis that the means are<a name="line.537"></a>
<FONT color="green">538</FONT>         * equal can be rejected with confidence &lt;code&gt;1 - alpha&lt;/code&gt;.  To <a name="line.538"></a>
<FONT color="green">539</FONT>         * perform a 1-sided test, use &lt;code&gt;alpha * 2&lt;/code&gt;&lt;/p&gt;<a name="line.539"></a>
<FONT color="green">540</FONT>         * &lt;p&gt;<a name="line.540"></a>
<FONT color="green">541</FONT>         * See {@link #t(double[], double[])} for the formula used to compute the<a name="line.541"></a>
<FONT color="green">542</FONT>         * t-statistic.  Degrees of freedom are approximated using the<a name="line.542"></a>
<FONT color="green">543</FONT>         * &lt;a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm"&gt;<a name="line.543"></a>
<FONT color="green">544</FONT>         * Welch-Satterthwaite approximation.&lt;/a&gt;&lt;/p&gt;<a name="line.544"></a>
<FONT color="green">545</FONT>         * &lt;p&gt;<a name="line.545"></a>
<FONT color="green">546</FONT>         * &lt;strong&gt;Examples:&lt;/strong&gt;&lt;br&gt;&lt;ol&gt;<a name="line.546"></a>
<FONT color="green">547</FONT>         * &lt;li&gt;To test the (2-sided) hypothesis &lt;code&gt;mean 1 = mean 2 &lt;/code&gt; at<a name="line.547"></a>
<FONT color="green">548</FONT>         * the 95% level,  use <a name="line.548"></a>
<FONT color="green">549</FONT>         * &lt;br&gt;&lt;code&gt;tTest(sample1, sample2, 0.05). &lt;/code&gt;<a name="line.549"></a>
<FONT color="green">550</FONT>         * &lt;/li&gt;<a name="line.550"></a>
<FONT color="green">551</FONT>         * &lt;li&gt;To test the (one-sided) hypothesis &lt;code&gt; mean 1 &lt; mean 2 &lt;/code&gt;,<a name="line.551"></a>
<FONT color="green">552</FONT>         * at the 99% level, first verify that the measured  mean of &lt;code&gt;sample 1&lt;/code&gt;<a name="line.552"></a>
<FONT color="green">553</FONT>         * is less than the mean of &lt;code&gt;sample 2&lt;/code&gt; and then use <a name="line.553"></a>
<FONT color="green">554</FONT>         * &lt;br&gt;&lt;code&gt;tTest(sample1, sample2, 0.02) &lt;/code&gt;<a name="line.554"></a>
<FONT color="green">555</FONT>         * &lt;/li&gt;&lt;/ol&gt;&lt;/p&gt;<a name="line.555"></a>
<FONT color="green">556</FONT>         * &lt;p&gt;<a name="line.556"></a>
<FONT color="green">557</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.557"></a>
<FONT color="green">558</FONT>         * The validity of the test depends on the assumptions of the parametric<a name="line.558"></a>
<FONT color="green">559</FONT>         * t-test procedure, as discussed <a name="line.559"></a>
<FONT color="green">560</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;<a name="line.560"></a>
<FONT color="green">561</FONT>         * here&lt;/a&gt;&lt;/p&gt;<a name="line.561"></a>
<FONT color="green">562</FONT>         * &lt;p&gt;<a name="line.562"></a>
<FONT color="green">563</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.563"></a>
<FONT color="green">564</FONT>         * &lt;li&gt;The observed array lengths must both be at least 2.<a name="line.564"></a>
<FONT color="green">565</FONT>         * &lt;/li&gt;<a name="line.565"></a>
<FONT color="green">566</FONT>         * &lt;li&gt; &lt;code&gt; 0 &lt; alpha &lt; 0.5 &lt;/code&gt;<a name="line.566"></a>
<FONT color="green">567</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.567"></a>
<FONT color="green">568</FONT>         *<a name="line.568"></a>
<FONT color="green">569</FONT>         * @param sample1 array of sample data values<a name="line.569"></a>
<FONT color="green">570</FONT>         * @param sample2 array of sample data values<a name="line.570"></a>
<FONT color="green">571</FONT>         * @param alpha significance level of the test<a name="line.571"></a>
<FONT color="green">572</FONT>         * @return true if the null hypothesis can be rejected with <a name="line.572"></a>
<FONT color="green">573</FONT>         * confidence 1 - alpha<a name="line.573"></a>
<FONT color="green">574</FONT>         * @throws IllegalArgumentException if the preconditions are not met<a name="line.574"></a>
<FONT color="green">575</FONT>         * @throws MathException if an error occurs performing the test<a name="line.575"></a>
<FONT color="green">576</FONT>         */<a name="line.576"></a>
<FONT color="green">577</FONT>        public abstract boolean tTest(<a name="line.577"></a>
<FONT color="green">578</FONT>            double[] sample1,<a name="line.578"></a>
<FONT color="green">579</FONT>            double[] sample2,<a name="line.579"></a>
<FONT color="green">580</FONT>            double alpha)<a name="line.580"></a>
<FONT color="green">581</FONT>            throws IllegalArgumentException, MathException;<a name="line.581"></a>
<FONT color="green">582</FONT>        /**<a name="line.582"></a>
<FONT color="green">583</FONT>         * Performs a <a name="line.583"></a>
<FONT color="green">584</FONT>         * &lt;a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"&gt;<a name="line.584"></a>
<FONT color="green">585</FONT>         * two-sided t-test&lt;/a&gt; evaluating the null hypothesis that &lt;code&gt;sample1&lt;/code&gt; <a name="line.585"></a>
<FONT color="green">586</FONT>         * and &lt;code&gt;sample2&lt;/code&gt; are drawn from populations with the same mean, <a name="line.586"></a>
<FONT color="green">587</FONT>         * with significance level &lt;code&gt;alpha&lt;/code&gt;,  assuming that the<a name="line.587"></a>
<FONT color="green">588</FONT>         * subpopulation variances are equal.  Use <a name="line.588"></a>
<FONT color="green">589</FONT>         * {@link #tTest(double[], double[], double)} to perform the test without<a name="line.589"></a>
<FONT color="green">590</FONT>         * the assumption of equal variances.<a name="line.590"></a>
<FONT color="green">591</FONT>         * &lt;p&gt;<a name="line.591"></a>
<FONT color="green">592</FONT>         * Returns &lt;code&gt;true&lt;/code&gt; iff the null hypothesis that the means are<a name="line.592"></a>
<FONT color="green">593</FONT>         * equal can be rejected with confidence &lt;code&gt;1 - alpha&lt;/code&gt;.  To <a name="line.593"></a>
<FONT color="green">594</FONT>         * perform a 1-sided test, use &lt;code&gt;alpha * 2.&lt;/code&gt;  To perform the test<a name="line.594"></a>
<FONT color="green">595</FONT>         * without the assumption of equal subpopulation variances, use <a name="line.595"></a>
<FONT color="green">596</FONT>         * {@link #tTest(double[], double[], double)}.&lt;/p&gt;<a name="line.596"></a>
<FONT color="green">597</FONT>         * &lt;p&gt;<a name="line.597"></a>
<FONT color="green">598</FONT>         * A pooled variance estimate is used to compute the t-statistic. See<a name="line.598"></a>
<FONT color="green">599</FONT>         * {@link #t(double[], double[])} for the formula. The sum of the sample<a name="line.599"></a>
<FONT color="green">600</FONT>         * sizes minus 2 is used as the degrees of freedom.&lt;/p&gt;<a name="line.600"></a>
<FONT color="green">601</FONT>         * &lt;p&gt;<a name="line.601"></a>
<FONT color="green">602</FONT>         * &lt;strong&gt;Examples:&lt;/strong&gt;&lt;br&gt;&lt;ol&gt;<a name="line.602"></a>
<FONT color="green">603</FONT>         * &lt;li&gt;To test the (2-sided) hypothesis &lt;code&gt;mean 1 = mean 2 &lt;/code&gt; at<a name="line.603"></a>
<FONT color="green">604</FONT>         * the 95% level, use &lt;br&gt;&lt;code&gt;tTest(sample1, sample2, 0.05). &lt;/code&gt;<a name="line.604"></a>
<FONT color="green">605</FONT>         * &lt;/li&gt;<a name="line.605"></a>
<FONT color="green">606</FONT>         * &lt;li&gt;To test the (one-sided) hypothesis &lt;code&gt; mean 1 &lt; mean 2, &lt;/code&gt;<a name="line.606"></a>
<FONT color="green">607</FONT>         * at the 99% level, first verify that the measured mean of <a name="line.607"></a>
<FONT color="green">608</FONT>         * &lt;code&gt;sample 1&lt;/code&gt; is less than the mean of &lt;code&gt;sample 2&lt;/code&gt;<a name="line.608"></a>
<FONT color="green">609</FONT>         * and then use<a name="line.609"></a>
<FONT color="green">610</FONT>         * &lt;br&gt;&lt;code&gt;tTest(sample1, sample2, 0.02) &lt;/code&gt;<a name="line.610"></a>
<FONT color="green">611</FONT>         * &lt;/li&gt;&lt;/ol&gt;&lt;/p&gt;<a name="line.611"></a>
<FONT color="green">612</FONT>         * &lt;p&gt;<a name="line.612"></a>
<FONT color="green">613</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.613"></a>
<FONT color="green">614</FONT>         * The validity of the test depends on the assumptions of the parametric<a name="line.614"></a>
<FONT color="green">615</FONT>         * t-test procedure, as discussed <a name="line.615"></a>
<FONT color="green">616</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;<a name="line.616"></a>
<FONT color="green">617</FONT>         * here&lt;/a&gt;&lt;/p&gt;<a name="line.617"></a>
<FONT color="green">618</FONT>         * &lt;p&gt;<a name="line.618"></a>
<FONT color="green">619</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.619"></a>
<FONT color="green">620</FONT>         * &lt;li&gt;The observed array lengths must both be at least 2.<a name="line.620"></a>
<FONT color="green">621</FONT>         * &lt;/li&gt;<a name="line.621"></a>
<FONT color="green">622</FONT>         * &lt;li&gt; &lt;code&gt; 0 &lt; alpha &lt; 0.5 &lt;/code&gt;<a name="line.622"></a>
<FONT color="green">623</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.623"></a>
<FONT color="green">624</FONT>         *<a name="line.624"></a>
<FONT color="green">625</FONT>         * @param sample1 array of sample data values<a name="line.625"></a>
<FONT color="green">626</FONT>         * @param sample2 array of sample data values<a name="line.626"></a>
<FONT color="green">627</FONT>         * @param alpha significance level of the test<a name="line.627"></a>
<FONT color="green">628</FONT>         * @return true if the null hypothesis can be rejected with <a name="line.628"></a>
<FONT color="green">629</FONT>         * confidence 1 - alpha<a name="line.629"></a>
<FONT color="green">630</FONT>         * @throws IllegalArgumentException if the preconditions are not met<a name="line.630"></a>
<FONT color="green">631</FONT>         * @throws MathException if an error occurs performing the test<a name="line.631"></a>
<FONT color="green">632</FONT>         */<a name="line.632"></a>
<FONT color="green">633</FONT>        public abstract boolean homoscedasticTTest(<a name="line.633"></a>
<FONT color="green">634</FONT>            double[] sample1,<a name="line.634"></a>
<FONT color="green">635</FONT>            double[] sample2,<a name="line.635"></a>
<FONT color="green">636</FONT>            double alpha)<a name="line.636"></a>
<FONT color="green">637</FONT>            throws IllegalArgumentException, MathException;<a name="line.637"></a>
<FONT color="green">638</FONT>        /**<a name="line.638"></a>
<FONT color="green">639</FONT>         * Returns the &lt;i&gt;observed significance level&lt;/i&gt;, or <a name="line.639"></a>
<FONT color="green">640</FONT>         * &lt;i&gt;p-value&lt;/i&gt;, associated with a two-sample, two-tailed t-test <a name="line.640"></a>
<FONT color="green">641</FONT>         * comparing the means of the datasets described by two StatisticalSummary<a name="line.641"></a>
<FONT color="green">642</FONT>         * instances.<a name="line.642"></a>
<FONT color="green">643</FONT>         * &lt;p&gt;<a name="line.643"></a>
<FONT color="green">644</FONT>         * The number returned is the smallest significance level<a name="line.644"></a>
<FONT color="green">645</FONT>         * at which one can reject the null hypothesis that the two means are<a name="line.645"></a>
<FONT color="green">646</FONT>         * equal in favor of the two-sided alternative that they are different. <a name="line.646"></a>
<FONT color="green">647</FONT>         * For a one-sided test, divide the returned value by 2.&lt;/p&gt;<a name="line.647"></a>
<FONT color="green">648</FONT>         * &lt;p&gt;<a name="line.648"></a>
<FONT color="green">649</FONT>         * The test does not assume that the underlying popuation variances are<a name="line.649"></a>
<FONT color="green">650</FONT>         * equal  and it uses approximated degrees of freedom computed from the <a name="line.650"></a>
<FONT color="green">651</FONT>         * sample data to compute the p-value.   To perform the test assuming<a name="line.651"></a>
<FONT color="green">652</FONT>         * equal variances, use <a name="line.652"></a>
<FONT color="green">653</FONT>         * {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.&lt;/p&gt;<a name="line.653"></a>
<FONT color="green">654</FONT>         * &lt;p&gt;<a name="line.654"></a>
<FONT color="green">655</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.655"></a>
<FONT color="green">656</FONT>         * The validity of the p-value depends on the assumptions of the parametric<a name="line.656"></a>
<FONT color="green">657</FONT>         * t-test procedure, as discussed <a name="line.657"></a>
<FONT color="green">658</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;<a name="line.658"></a>
<FONT color="green">659</FONT>         * here&lt;/a&gt;&lt;/p&gt;<a name="line.659"></a>
<FONT color="green">660</FONT>         * &lt;p&gt;<a name="line.660"></a>
<FONT color="green">661</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.661"></a>
<FONT color="green">662</FONT>         * &lt;li&gt;The datasets described by the two Univariates must each contain<a name="line.662"></a>
<FONT color="green">663</FONT>         * at least 2 observations.<a name="line.663"></a>
<FONT color="green">664</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.664"></a>
<FONT color="green">665</FONT>         *<a name="line.665"></a>
<FONT color="green">666</FONT>         * @param sampleStats1  StatisticalSummary describing data from the first sample<a name="line.666"></a>
<FONT color="green">667</FONT>         * @param sampleStats2  StatisticalSummary describing data from the second sample<a name="line.667"></a>
<FONT color="green">668</FONT>         * @return p-value for t-test<a name="line.668"></a>
<FONT color="green">669</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.669"></a>
<FONT color="green">670</FONT>         * @throws MathException if an error occurs computing the p-value<a name="line.670"></a>
<FONT color="green">671</FONT>         */<a name="line.671"></a>
<FONT color="green">672</FONT>        public abstract double tTest(<a name="line.672"></a>
<FONT color="green">673</FONT>            StatisticalSummary sampleStats1,<a name="line.673"></a>
<FONT color="green">674</FONT>            StatisticalSummary sampleStats2)<a name="line.674"></a>
<FONT color="green">675</FONT>            throws IllegalArgumentException, MathException;<a name="line.675"></a>
<FONT color="green">676</FONT>        /**<a name="line.676"></a>
<FONT color="green">677</FONT>         * Returns the &lt;i&gt;observed significance level&lt;/i&gt;, or <a name="line.677"></a>
<FONT color="green">678</FONT>         * &lt;i&gt;p-value&lt;/i&gt;, associated with a two-sample, two-tailed t-test <a name="line.678"></a>
<FONT color="green">679</FONT>         * comparing the means of the datasets described by two StatisticalSummary<a name="line.679"></a>
<FONT color="green">680</FONT>         * instances, under the hypothesis of equal subpopulation variances. To<a name="line.680"></a>
<FONT color="green">681</FONT>         * perform a test without the equal variances assumption, use<a name="line.681"></a>
<FONT color="green">682</FONT>         * {@link #tTest(StatisticalSummary, StatisticalSummary)}.<a name="line.682"></a>
<FONT color="green">683</FONT>         * &lt;p&gt;<a name="line.683"></a>
<FONT color="green">684</FONT>         * The number returned is the smallest significance level<a name="line.684"></a>
<FONT color="green">685</FONT>         * at which one can reject the null hypothesis that the two means are<a name="line.685"></a>
<FONT color="green">686</FONT>         * equal in favor of the two-sided alternative that they are different. <a name="line.686"></a>
<FONT color="green">687</FONT>         * For a one-sided test, divide the returned value by 2.&lt;/p&gt;<a name="line.687"></a>
<FONT color="green">688</FONT>         * &lt;p&gt;<a name="line.688"></a>
<FONT color="green">689</FONT>         * See {@link #homoscedasticT(double[], double[])} for the formula used to<a name="line.689"></a>
<FONT color="green">690</FONT>         * compute the t-statistic. The sum of the  sample sizes minus 2 is used as<a name="line.690"></a>
<FONT color="green">691</FONT>         * the degrees of freedom.&lt;/p&gt;<a name="line.691"></a>
<FONT color="green">692</FONT>         * &lt;p&gt;<a name="line.692"></a>
<FONT color="green">693</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.693"></a>
<FONT color="green">694</FONT>         * The validity of the p-value depends on the assumptions of the parametric<a name="line.694"></a>
<FONT color="green">695</FONT>         * t-test procedure, as discussed <a name="line.695"></a>
<FONT color="green">696</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;here&lt;/a&gt;<a name="line.696"></a>
<FONT color="green">697</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.697"></a>
<FONT color="green">698</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.698"></a>
<FONT color="green">699</FONT>         * &lt;li&gt;The datasets described by the two Univariates must each contain<a name="line.699"></a>
<FONT color="green">700</FONT>         * at least 2 observations.<a name="line.700"></a>
<FONT color="green">701</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.701"></a>
<FONT color="green">702</FONT>         *<a name="line.702"></a>
<FONT color="green">703</FONT>         * @param sampleStats1  StatisticalSummary describing data from the first sample<a name="line.703"></a>
<FONT color="green">704</FONT>         * @param sampleStats2  StatisticalSummary describing data from the second sample<a name="line.704"></a>
<FONT color="green">705</FONT>         * @return p-value for t-test<a name="line.705"></a>
<FONT color="green">706</FONT>         * @throws IllegalArgumentException if the precondition is not met<a name="line.706"></a>
<FONT color="green">707</FONT>         * @throws MathException if an error occurs computing the p-value<a name="line.707"></a>
<FONT color="green">708</FONT>         */<a name="line.708"></a>
<FONT color="green">709</FONT>        public abstract double homoscedasticTTest(<a name="line.709"></a>
<FONT color="green">710</FONT>            StatisticalSummary sampleStats1,<a name="line.710"></a>
<FONT color="green">711</FONT>            StatisticalSummary sampleStats2)<a name="line.711"></a>
<FONT color="green">712</FONT>            throws IllegalArgumentException, MathException;<a name="line.712"></a>
<FONT color="green">713</FONT>        /**<a name="line.713"></a>
<FONT color="green">714</FONT>         * Performs a <a name="line.714"></a>
<FONT color="green">715</FONT>         * &lt;a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"&gt;<a name="line.715"></a>
<FONT color="green">716</FONT>         * two-sided t-test&lt;/a&gt; evaluating the null hypothesis that <a name="line.716"></a>
<FONT color="green">717</FONT>         * &lt;code&gt;sampleStats1&lt;/code&gt; and &lt;code&gt;sampleStats2&lt;/code&gt; describe<a name="line.717"></a>
<FONT color="green">718</FONT>         * datasets drawn from populations with the same mean, with significance<a name="line.718"></a>
<FONT color="green">719</FONT>         * level &lt;code&gt;alpha&lt;/code&gt;.   This test does not assume that the<a name="line.719"></a>
<FONT color="green">720</FONT>         * subpopulation variances are equal.  To perform the test under the equal<a name="line.720"></a>
<FONT color="green">721</FONT>         * variances assumption, use<a name="line.721"></a>
<FONT color="green">722</FONT>         * {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.<a name="line.722"></a>
<FONT color="green">723</FONT>         * &lt;p&gt;<a name="line.723"></a>
<FONT color="green">724</FONT>         * Returns &lt;code&gt;true&lt;/code&gt; iff the null hypothesis that the means are<a name="line.724"></a>
<FONT color="green">725</FONT>         * equal can be rejected with confidence &lt;code&gt;1 - alpha&lt;/code&gt;.  To <a name="line.725"></a>
<FONT color="green">726</FONT>         * perform a 1-sided test, use &lt;code&gt;alpha * 2&lt;/code&gt;&lt;/p&gt;<a name="line.726"></a>
<FONT color="green">727</FONT>         * &lt;p&gt;<a name="line.727"></a>
<FONT color="green">728</FONT>         * See {@link #t(double[], double[])} for the formula used to compute the<a name="line.728"></a>
<FONT color="green">729</FONT>         * t-statistic.  Degrees of freedom are approximated using the<a name="line.729"></a>
<FONT color="green">730</FONT>         * &lt;a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm"&gt;<a name="line.730"></a>
<FONT color="green">731</FONT>         * Welch-Satterthwaite approximation.&lt;/a&gt;&lt;/p&gt;<a name="line.731"></a>
<FONT color="green">732</FONT>         * &lt;p&gt;<a name="line.732"></a>
<FONT color="green">733</FONT>         * &lt;strong&gt;Examples:&lt;/strong&gt;&lt;br&gt;&lt;ol&gt;<a name="line.733"></a>
<FONT color="green">734</FONT>         * &lt;li&gt;To test the (2-sided) hypothesis &lt;code&gt;mean 1 = mean 2 &lt;/code&gt; at<a name="line.734"></a>
<FONT color="green">735</FONT>         * the 95%, use <a name="line.735"></a>
<FONT color="green">736</FONT>         * &lt;br&gt;&lt;code&gt;tTest(sampleStats1, sampleStats2, 0.05) &lt;/code&gt;<a name="line.736"></a>
<FONT color="green">737</FONT>         * &lt;/li&gt;<a name="line.737"></a>
<FONT color="green">738</FONT>         * &lt;li&gt;To test the (one-sided) hypothesis &lt;code&gt; mean 1 &lt; mean 2 &lt;/code&gt;<a name="line.738"></a>
<FONT color="green">739</FONT>         * at the 99% level,  first verify that the measured mean of  <a name="line.739"></a>
<FONT color="green">740</FONT>         * &lt;code&gt;sample 1&lt;/code&gt; is less than  the mean of &lt;code&gt;sample 2&lt;/code&gt;<a name="line.740"></a>
<FONT color="green">741</FONT>         * and then use <a name="line.741"></a>
<FONT color="green">742</FONT>         * &lt;br&gt;&lt;code&gt;tTest(sampleStats1, sampleStats2, 0.02) &lt;/code&gt;<a name="line.742"></a>
<FONT color="green">743</FONT>         * &lt;/li&gt;&lt;/ol&gt;&lt;/p&gt;<a name="line.743"></a>
<FONT color="green">744</FONT>         * &lt;p&gt;<a name="line.744"></a>
<FONT color="green">745</FONT>         * &lt;strong&gt;Usage Note:&lt;/strong&gt;&lt;br&gt;<a name="line.745"></a>
<FONT color="green">746</FONT>         * The validity of the test depends on the assumptions of the parametric<a name="line.746"></a>
<FONT color="green">747</FONT>         * t-test procedure, as discussed <a name="line.747"></a>
<FONT color="green">748</FONT>         * &lt;a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"&gt;<a name="line.748"></a>
<FONT color="green">749</FONT>         * here&lt;/a&gt;&lt;/p&gt;<a name="line.749"></a>
<FONT color="green">750</FONT>         * &lt;p&gt;<a name="line.750"></a>
<FONT color="green">751</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;: &lt;ul&gt;<a name="line.751"></a>
<FONT color="green">752</FONT>         * &lt;li&gt;The datasets described by the two Univariates must each contain<a name="line.752"></a>
<FONT color="green">753</FONT>         * at least 2 observations.<a name="line.753"></a>
<FONT color="green">754</FONT>         * &lt;/li&gt;<a name="line.754"></a>
<FONT color="green">755</FONT>         * &lt;li&gt; &lt;code&gt; 0 &lt; alpha &lt; 0.5 &lt;/code&gt;<a name="line.755"></a>
<FONT color="green">756</FONT>         * &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.756"></a>
<FONT color="green">757</FONT>         *<a name="line.757"></a>
<FONT color="green">758</FONT>         * @param sampleStats1 StatisticalSummary describing sample data values<a name="line.758"></a>
<FONT color="green">759</FONT>         * @param sampleStats2 StatisticalSummary describing sample data values<a name="line.759"></a>
<FONT color="green">760</FONT>         * @param alpha significance level of the test<a name="line.760"></a>
<FONT color="green">761</FONT>         * @return true if the null hypothesis can be rejected with <a name="line.761"></a>
<FONT color="green">762</FONT>         * confidence 1 - alpha<a name="line.762"></a>
<FONT color="green">763</FONT>         * @throws IllegalArgumentException if the preconditions are not met<a name="line.763"></a>
<FONT color="green">764</FONT>         * @throws MathException if an error occurs performing the test<a name="line.764"></a>
<FONT color="green">765</FONT>         */<a name="line.765"></a>
<FONT color="green">766</FONT>        public abstract boolean tTest(<a name="line.766"></a>
<FONT color="green">767</FONT>            StatisticalSummary sampleStats1,<a name="line.767"></a>
<FONT color="green">768</FONT>            StatisticalSummary sampleStats2,<a name="line.768"></a>
<FONT color="green">769</FONT>            double alpha)<a name="line.769"></a>
<FONT color="green">770</FONT>            throws IllegalArgumentException, MathException;<a name="line.770"></a>
<FONT color="green">771</FONT>    }<a name="line.771"></a>




























































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